Let M be the coefficient of induction between the galvanometer coil and the suspended magnet. It is of the form M = G191 Q1 (0) + G2I2 Q2 (0) + &c., (4) where G1, G2, &c. are coefficients belonging to the coil, 91,92, &c. to the magnet, and Q1 (0), Q2 (0), &c., are zonal harmonics of the angle between the axes of the coil and the magnet. See Art. 700. By a proper arrangement of the coils of the galvanometer, and by building up the suspended magnet of several magnets placed side by side at proper distances, we may cause all the terms of M after the first to become insensible compared with the first. If we also put where G is the principal coefficient of the galvanometer, m is the magnetic moment of the magnet, and p is the angle between the axis of the magnet and the plane of the coil, which, in this experiment, is always a small angle. If L is the coefficient of self-induction of the coil, and R its resistance, and y the current in the coil, The moment of the force with which the current y acts on the dM magnet is y , or Gmy cosp. The angle & is in this experiment do Let us suppose that the equation of motion of the magnet when the circuit is broken is where A is the moment of inertia of the suspended apparatus, B dt expresses the resistance arising from the viscosity of the air and of the suspension fibre, &c., and Co expresses the moment of the force arising from the earth's magnetism, the torsion of the suspension apparatus, &c., tending to bring the magnet to its position of equilibrium. The equation of motion, as affected by the current, will be 762.] WEBER'S METHOD. 363 To determine the motion of the magnet, we have to combine this equation with (7) and eliminate y. The result is a linear differential equation of the third order. We have no occasion, however, to solve this equation, because the data of the problem are the observed elements of the motion of the magnet, and from these we have to determine the value of R. Let and πο be the values of a and w in equation (2) when the circuit is broken. In this case R is infinite, and the equation is reduced to the form (8). We thus find B = 2 Aα, C = A (a2 + w2). Solving equation (10) for R, and writing d (a+iw), where i=√1, a+iw a2-w2+2iaw-2a (a+iw)+a2 + wo2 (11) (12) 2 + L(a+iw). (13) Since the value of ∞ is in general much greater than that of a, the best value of R is found by equating the terms in iw, We may also obtain a value of R by equating the terms not involving i, but as these terms are small, the equation is useful only as a means of testing the accuracy of the observations. From these equations we find the following testing equation, = LA {(a− a)1 + 2 (a− a)2 (w2+w2)+(w2 − ∞ 2)2}. (15) Since LA2 is very small compared with G2m2, this equation gives In this expression G may be determined either from the linear measurement of the galvanometer coil, or better, by comparison with a standard coil, according to the method of Art. 753. 4 is the moment of inertia of the magnet and its suspended apparatus, which is to be found by the proper dynamical method. w, wo, a and a, are given by observation. The determination of the value of m, the magnetic moment of the suspended magnet, is the most difficult part of the investigation, because it is affected by temperature, by the earth's magnetic force, and by mechanical violence, so that great care must be taken to measure this quantity when the magnet is in the very same circumstances as when it is vibrating. The second term of R, that which involves L, is of less importance, as it is generally small compared with the first term. The value of L may be determined either by calculation from the known form of the coil, or by an experiment on the extra-current of induction. See Art. 756. Thomson's Method by a Revolving Coil. 763.] This method was suggested by Thomson to the Committee. of the British Association on Electrical Standards, and the experiment was made by M. M. Balfour Stewart, Fleeming Jenkin, and the author in 1863*. A circular coil is made to revolve with uniform velocity about a vertical axis. A small magnet is suspended by a silk fibre at the centre of the coil. An electric current is induced in the coil by the earth's magnetism, and also by the suspended magnet. This current is periodic, flowing in opposite directions through the wire of the coil during different parts of each revolution, but the effect of the current on the suspended magnet is to produce a deflexion from the magnetic meridian in the direction of the rotation of the coil. 764.] Let I be the horizontal component of the earth's magnetism. Let y be the strength of the current in the coil. the total area inclosed by all the windings of the wire. G the magnetic force at the centre of the coil due to unit current. L the coefficient of self-induction of the coil. M the magnetic moment of the suspended magnet. the angle between the plane of the coil and the magnetic meridian. the angle between the axis of the suspended magnet and the magnetic meridian A the moment of inertia of the suspended magnet. MHT the coefficient of torsion of the suspension fibre. * See Report of the British Association for 1863. 765.] THOMSON'S METHOD. The kinetic energy of the system is 365 T = { Ly2 — Hgy sin 0-M Gy sin (0—4) + MH cos &+ § A¿2. (1) The first term, Ly2, expresses the energy of the current as depending on the coil itself. The second term depends on the mutual action of the current and terrestrial magnetism, the third on that of the current and the magnetism of the suspended magnet, the fourth on that of the magnetism of the suspended magnet and terrestrial magnetism, and the last expresses the kinetic energy of the matter composing the magnet and the suspended apparatus which moves with it. The potential energy of the suspended apparatus arising from the torsion of the fibre is and if R is the resistance of the coil, the equation of the current is or, since d dt R+L ) y = Hgw cos 0 + MG (w− p) cos (0 − p). (4) (5) (6) 765.] It is the result alike of theory and observation that p, the azimuth of the magnet, is subject to two kinds of periodic variations. One of these is a free oscillation, whose periodic time depends on the intensity of terrestrial magnetism, and is, in the experiment, several seconds. The other is a forced vibration whose period is half that of the revolving coil, and whose amplitude is, as we shall see, insensible. Hence, in determining y, we may treat as sensibly constant. The last term of this expression soon dies away when the rotation is continued uniform. whence 4-MGy cos (0-4)+ MH (sin +7 (-a)) = 0. (11) Substituting the value of y, and arranging the terms according to the functions of multiples of 0, then we know from observation that where is the mean value of 4, and the second term expresses the free vibrations gradually decaying, and the third the forced vibrations arising from the variation of the deflecting current. coil makes many revolutions during one free vibration of the magnet, the amplitude of the forced vibrations of the magnet is very small, and we may neglect the terms in (11) which involve c. Beginning with the terms in (11) which do not involve 0, we find MHGgw R2 + L2 w2 (R cos 40+ Lw sin 。)+ M2G2 (w-p) R R2 + L2 (w−4)2 = MH (sin po+T (%-a)). (13) Remembering that is small, and that I is generally small compared with Gg, we find as a sufficiently approximate value of R, 766.] The resistance is thus determined in electromagnetic measure in terms of the velocity w and the deviation . It is not necessary to determine H, the horizontal terrestrial magnetic force, provided it remains constant during the experiment. M To determine we must make use of the suspended magnet to deflect the magnet of the magnetometer, as described in Art. 454. In this experiment M should be small, so that this correction becomes of secondary importance. For the other corrections required in this experiment see the Report of the British Association for 1863, p. 168. |